
The standard polynomial \(s_2\)  itself is a Lie
polynomial.

\(s_4\)  vanishes on \(\operatorname{sl}_2\)  (viewed inside the associative algebra \(M_2(K)\) ),
since \(\operatorname{sl}_2\)  has dimension 3. But surprisingly, this is not the
polynomial of lowest degree vanishing on \(\operatorname{sl}_2\) , as we see next.

Bakhturin [1]} points out that \(f = [(x_1x_2 +x_2 x_1),x_3]\)  vanishes on \(\operatorname{sl}_2\) . In other words, \(a_1a_2 + a_2a_1\)  is scalar for any \(2 \times 2\)  matrices \(a_1,a_2\)  of trace 0.
Indeed, \(a_i^2 \)  is scalar for \(i=1,2\) , implying \(a_1a_2 + a_2 a_1\) 
is scalar unless \(a_1\) , \(a_2\)  are linearly independent, in which case
\(a_1a_2 + a_2 a_1\)  commutes with both \(a_1 \)  and \( a_2\) , and thus
again is scalar. But \(f\)  is not a Lie polynomial, as seen via the
next lemma.


